Adaptive Reduction of Design Variables Using Global Sensitivity in Reliability-Based Optimization

Transcription

1 Adaptve Reducton of Desgn Varables Usng Global Senstvty n Relablty-Based Optmzaton Nam H. Km * and Haoyu Wang Dept. of Mechancal & Aerospace Engneerng, Unversty of Florda, Ganesvlle, Florda, 326 Nestor V. Quepo Dept. of Mechancal & Aerospace Engneerng, Unversty of Florda, Ganesvlle, Florda, 326 Appled Computng Insttute, Faculty of Engneerng, Unversty of Zula, Venezuela Ths paper presents an effcent shape optmzaton technque based on stochastc response surfaces (SRS) and adaptve reducton of random varables usng global senstvty nformaton. Each SRS s a polynomal chaos expanson that uses Hermte polynomal bases and provdes a closed form soluton of the model output from a sgnfcant lower number of model smulatons than those requred by conventonal methods such as modfed Monte Carlo Methods and Latn Hypercube Samplng. Random varables are adaptvely fxed before constructng the SRS f ther correspondng global senstvty ndces calculated usng low-order SRS are below a certan threshold. Usng SRS and adaptve reducton of random varables, relablty-based optmzaton problems can be solved wth a reasonable amount of computatonal cost. The effcency and convergence of the proposed approach are demonstrated usng a benchmark case and an ndustral relablty-based desgn optmzaton problem (automotve part). Nomenclature u = vector of standard random varables x = vector of random varables d = vector of desgn parameters Γ p ( u,, up) = multdmensonal Hermte polynomals of degree p [K] = structural stffness matrx {F} = structural load vector {D} = nodal soluton vector (dsplacement) S = global senstvty ndex of -th random varable total S = total senstvty ndex of -th random varable E( ) = expected value V( ) = varance G p = approxmaton of the performance functon wth p-th order Hermte polynomals σ max = maxmum allowable equvalent stress c(d) = cost functon P f = falure probablty β t = target relablty Φ = cumulatve dstrbuton functon of the standard random varable * Assstant Professor, Member AIAA Graduate Assstant Vstng Professor Amercan Insttute of Aeronautcs and Astronautcs

2 U I. Introducton NCERTAINTY n the desgn parameters makes shape optmzaton of structural systems a computatonally expensve task due to the sgnfcant number of structural analyses requred by tradtonal methods. Crtcal ssues for overcomng these dffcultes are those related to uncertanty characterzaton, uncertanty propagaton, rankng of desgn varables, and effcent optmzaton algorthms. Tradtonal approaches for these tasks often fal to meet constrants (computatonal resources, cost, tme, etc.) typcally present n ndustral envronments. In partcular, relablty-based desgn optmzaton (RBDO) nvolvng a computatonally demandng model has been lmted by the relatvely hgh number of requred analyses for uncertanty propagaton durng the optmzaton process. Whle there has been progress addressng ths ssue, such as more effcent momentbased optmzaton algorthms (e.g. RIA, PMA ), and the constructon of stochastc response surfaces (SRS) for uncertanty propagaton, 2 the possblty of reducng the number of analyses by systematcally fxng unessental desgn varables throughout the optmzaton process has not been fully explored. In ths paper, n order to avod the shortcomngs of the conventonal moment-based methods (FORM or SORM) and modfed Monte Carlo Methods and Latn Hypercube Samplng, and those assocated wth the use of SRS: ) local senstvty nformaton at samplng ponts s also used, and ) global senstvty ndces are calculated to decde whether to fx random varables whose contrbuton to the output varablty s less than a certan threshold. Wth reference to Fgure, the proposed approach for RBDO ntally constructs a low-order SRS usng all varables, and adaptvely reduces them dependng on the values of ther correspondng Global Senstvty Indces (GSI). GSI are calculated usng a varancebased method 3,4,5 a rgorous and theoretcally sound approach for global senstvty. Usng the reduced number of random varables, a hgh-order SRS s constructed from whch the relablty of the performance functon s evaluated. The paper s structured as follows: Secton 2 descrbes the uncertanty characterzaton of model nputs, and the uncertanty propagaton to the output usng the SRS. Secton 3 presents the procedure to compute global senstvty ndces n order to fx unessental random varables durng the constructon of the SRS. An RBDO problem s formulated and the results obtaned usng the proposed approach are the subject of Secton 4, followed by numercal examples n Secton 5 Random Desgn Varables X X 2 X n Low-Order Stochastc Response Surface Global Senstvty X n Hgher-Order Stochastc Response Surface X X 2 X 3 X 4 Relablty-based Desgn Optmzaton Mn. Cost s.t. PG ( ( X) 0) Pf L U X X X Optmzed? Fxng Unessental Varables No Update Desgn II. Uncertanty Quantfcaton Uncertanty quantfcaton can be decomposed n three fundamental steps: ) uncertanty characterzaton of model nputs, ) propagaton of uncertanty, and ) uncertanty management/decson makng. To assst the latter step, global senstvty of model nput to outputs s also ncorporated. Stop Yes Fgure. Adaptve reducton of unessental random desgn varables usng global senstvty ndces n RBDO. Loworder SRS s used for global senstvty analyss, whle a hgher-order SRS s used to evaluate the relablty of the 2 Amercan Insttute of Aeronautcs and Astronautcs

3 The uncertanty n model nputs s represented n terms of standardzed random varables (SRV) wth mean zero and varance equal to one. The selecton s supported by the fact that they are wdely used and well-behaved. For other types of random varables, an approprate transformaton must be employed. We wll assume that the model nputs are ndependent so each one s expressed drectly as a functon of SRV through a proper transformaton. Devroye 6 presents the requred transformaton technques and approxmatons for a varety of probablty dstrbutons. More arbtrary probablty dstrbutons can be approxmated usng algebrac manpulatons or by seres expansons. The uncertanty propagaton s based on constructng stochastc response surfaces (polynomal chaos expanson). Stochastc response surfaces 7 can be vew as an extenson of classcal determnstc response surfaces for model outputs constructed usng uncertan nputs and performance data collected at heurstcally selected collocaton ponts. The polynomal expanson uses Hermte polynomal bases for the space of square-ntegrable probablty densty functons (PDF) and provdes a closed form soluton of model outputs from a sgnfcant lower number of model smulatons than those requred by conventonal methods such as modfed Monte Carlo Methods and Latn Hypercube Samplng. Let n be the number of random varables and p be the order of polynomal. The model output can then be expressed n terms of SRV u = {u, u 2,, u n } T as: n n n j p p p p p 0 j 2 j jk 3 j k = = j= = j= k= () G = a + a Γ ( u ) + a Γ ( u, u ) + a Γ ( u, u, u ) + where G p p p s the approxmated model output, the a, a, are determnstc coeffcents to be estmated, and the Γ ( u,, u ) are multdmensonal Hermte polynomals of degree p gven by: p p j p Γ p( u,, up) = ( ) e e u u p T T / 2u u / 2u u p (2) where u s the vector of p ndependent and dentcally dstrbuted normal random varables { u } p k k= that represent the model nput uncertantes. In general, the approxmaton accuracy ncreases wth the order of the polynomal and should be selected reflectng the accuracy needs and computatonal constrants. In addton, the approxmaton n Eq. () ncludes robust coeffcents hence exhbtng relatvely small changes from low to hgh-order approxmatons. The number of model smulatons requred to construct the SRS could be reduced when local senstvty nformaton s avalable. The ssue s how effcently the local senstvty nformaton can be calculated. If the global fnte dfference method s employed, there s no advantage n usng senstvty nformaton because each senstvty nformaton requres addtonal analyses. Recently, Isukapall et al. 8 used an automatc dfferentaton program (ADIFOR) to calculate the local senstvty of the model output wth respect to random varables and used them to construct a stochastc response surface. Ther results showed that local senstvty nformaton can sgnfcantly reduce the number of samplng ponts requred. However, the computatonal cost of the automatc dfferentaton s often hgher than that of drect analyss 9. In contrast, when the fnte element method s used, as dscussed by van Keulen et al. 0, desgn senstvty analyss can provde a very effcent tool for calculatng gradent nformaton because the senstvty equaton uses the same coeffcent matrx that s already factorzed from the orgnal analyss. In ths paper, the contnuum-based senstvty analyss s utlzed to calculate the gradent of the model output wth respect to random varables. In many fnte element-based structural analyses, the dscrete system s often represented usng a matrx equaton as [ K]{ D} = { F } (3) where [K] s the stffness matrx, {F} s the load vector, and {D} s the nodal soluton. The model output n Eq. () can be expressed as a functon of the nodal soluton. Thus, the local senstvty of the model output can be easly calculated f the local senstvty of the nodal soluton s avalable. When desgn parameters are defned, the matrx equaton (3) can be dfferentated wth respect to the desgn parameter d to obtan the followng desgn senstvty equaton: 3 Amercan Insttute of Aeronautcs and Astronautcs

4 D F K [ K] = { D } (4) d d d The above equaton can be solved nexpensvely because the matrx [K] s already factorzed when solvng Eq. (3). The computatonal cost of senstvty analyss s less than 20% of the orgnal analyss cost. The effcency of the uncertanty propagaton approach s crtcal to RBDO (uncertanty management) snce at each desgn cycle an updated verson of the PDF for the constrant functon (related to model outputs) s requred. III. Global Senstvty Indces and an Adaptve Approach for Fxng Unessental Varables To reduce the number of smulatons requred to construct the SRS even further, unessental random varables are fxed durng the constructon of the SRS. A random varable s consdered unessental (and hence t s fxed) f ts contrbuton to the varance of the model output s below a gven threshold. Global senstvty ndces are calculated to quantfy the model nput contrbutons to the output varablty hence establshng whch factors nfluence the model predcton the most so that: ) resources can be focused to reduce or account for uncertanty where t s most approprate, or ) unessental varables can be fxed wthout sgnfcantly affectng the output varablty. The latter applcaton s the one of nterest n the context of ths work. Varance-based methods are the most rgorous and theoretcally sound approaches for total senstvty calculatons. 3,4,5 Varance based methods decompose the output varance nto partal varances of ncreasng dmensonalty as f( x ) = f + f ( x ) + f ( x, x ) + + f ( x, x,, x ) (5) 0 j j 2... n 2 n < j subject to the restrcton that: f 0 dx = for k,, s k s = (6) Specfcally, the global senstvty ndex S (man factor) and total senstvty ndex represented by Eqs. (7) and (8), respectvely: total S assocated wth x are * V( E( f x = x )) S = (7) V( f) S * total EV ( ( f x = x )) = (8) V( f) where E and V denote expected value and varance, respectvely. The symbol refers to all nput varables except x. Durng the desgn cycles varables wll be fxed based on the values of the global senstvty ndces (man factors) hence reducng the number of functon evaluatons requred for the constructon of the SRS. IV. Relablty-Based Optmzaton In order to llustrate and evaluate the proposed approach, a smple formulaton of the more general RBDO problem -4 s dscussed. The cost functon s assumed to be easly evaluated usng the desgn varables and the constrants are defned usng probablstc dstrbutons of the performance functons. Specfcally, consder the followng form of the RBDO problem: mnmze c( d) subject to PG ( j( x) < 0) Pf, j, j=,2,, np (9) d d d L U 4 Amercan Insttute of Aeronautcs and Astronautcs

5 where x = [x ] T ( =, 2,, n) denotes the vector of random parameters, d = [d ] T = [µ ] T represent the desgn varables chosen as the mean values of x, and c(d) dentfes the cost functon. The system performance crtera are descrbed by the performance functons G j (x) such that the system fals f G j (x) < 0. Each G j (x) s characterzed by ts cumulatve dstrbuton functon F G (g): F ( g) P( G ( x) g) f ( x) dx dx (0) = < = X Gj j n Gj ( x) < g where f X (x) s the jont PDF of all random system parameters and g s the probablstc performance measure. The relablty analyss of the performance functon requres evaluatng the non-decreasng F G (g) ~ g relatonshp, whch s performed n the probablty ntegraton doman bounded by the system parameter tolerance lmts. Snce the probablty ntegraton doman s n general complcated, many approxmaton methods (FORM or SORM) are often used. In ths paper, the PDF estmated usng the proposed uncertanty propagaton scheme s used for evaluatng relablty constrants hence provdng better approxmatons than tradtonal lnearzaton and thus sgnfcantly mprovng the rate of convergence of RBDO. Once the cost and constrant functons are evaluated, the optmzaton problem n Eq. (9) can be solved usng conventonal mathematcal programmng technques. V. Numercal Examples A. Stochastc Response Surface for the Torque-Arm Model Consder the torque arm model depcted n Fgure 2. 5 The locatons of boundary curves have uncertantes due to manufacturng processes. Thus, the relatve locatons of corner ponts of the boundary curves are defned as random varables. For smplcty, we assumed that all random varables exhbt a normal dstrbuton wth mean zero and standard devaton equal to 0.;.e., x~n(0, 0.). The mean values of these random varables are chosen as desgn parameters, whle, wthout loss of any generalty, the standard devaton remans constant durng the desgn process. As llustrated n Fgure 2, the ntal model conssts of eght desgn parameters. For example, desgn parameter d s the mean of the relatve locaton of pont A n the x-drecton. In order to show how the SRS s constructed and the PDF of the model output s calculated, we choose the three desgn parameters (d 2, d 6, and d 8 ) that most sgnfcantly contrbute to the stress performance at ponts A and B. A meshfree method 5 s employed to solve the structural response. In the ntal desgn, the maxmum stress of 305 MPa occurs at locaton A. For relablty analyss, the stress lmt s establshed to be 800MPa. In the relablty analyss the performance functon s defned such that G 0 s consdered falure. Thus, n the case of stress constrants, the followng performance functon s defned: G = σ max σ A A d 2 d d 6 d 5 Fgure 2. Shape desgn parameters for the torque arm model. Desgn parameters are the mean values of corner coordnates of boundary curves. Due to manufacturng processes, the ( x): ( x ) () B Symmetrc Desgn d 8 d 7 d 4 d N 2789N where σ max s the maxmum allowed equvalent stress and σ A s the stress at locaton A. Before constructng the stochastc response surface, t s mportant to transform the random varables {x (d, 0.)} nto standard normal dstrbutons {u }. After transformng to the standard normal dstrbutons, the stochastc response surface can be defned usng the polynomal chaos expanson. The 2nd- and 3rd-order Hermte polynomal chaos expansons can be wrtten as, respectvely, n n n n = ( ) + j j = = = j> (2) G a au a u auu and 5 Amercan Insttute of Aeronautcs and Astronautcs

6 n n = = G = a + a u + a ( u ) n n n a ( u 3 u ) ajuu j = = j> + + n n n 2 n n ajj ( uu j u ) ajkuu juk = j= = j> k> j + + (3) Note that the polynomals are constructed n the standard Gaussan space rather than the orgnal desgn space. For 2nd- and 3rd-order expanson, the numbers of unknown coeffcents, denoted by N 2 and N 3, are defned as N 2 = 0 and N 3 = 20. The coeffcents of the polynomal chaos expanson are obtaned usng the model outputs at selected collocaton ponts. The collocaton ponts are selected from the roots of the polynomal that s one order hgher than the polynomal chaos expanson. 6 For example, to solve for a three-dmensonal second order polynomal chaos expanson, the roots of the thrd order Hermte polynomal, 3, 0, and 3 are used, thus the possble collocaton ponts are (0, 0, 0), ( 3, 3, 3 ), ( 3, 0, 3 ), etc. There are 27 possble collocaton ponts and 0 unknown coeffcents n the case of second-order expanson. For robust estmaton of the regresson coeffcent, the number of collocaton ponts n general should be twce the number of unknown coeffcents. After choosng collocaton ponts n the standard normal space, a transformaton s appled from standard Gaussan space to desgn space accordng to the PDF assocated wth the desgn varables. In the torque arm model, the PDF of the performance functon s plotted n Fgure 3(a) for polynomals of dfferent orders. The accuracy and the convergence of the stochastc response surfaces are compared wth the PDF obtaned usng Monte Carlo smulaton wth 00,000 sample ponts. As expected, the root mean square error s reduced for hgher-order polynomals. In order to reduce the requred number of samplng ponts to construct the SRS, local senstvty nformaton s also used. At each samplng pont, n+ data are avalable (functon value + gradents of n random varables). In order to account for the local senstvty nformaton, the expressons n Eqs. (2) and (3) are dfferentated wth respect to the random varables. However, the stochastc response surfaces are defned n the standard Gaussan space. As a result, t s necessary to transform the local senstvty n the desgn space nto standard Gaussan space usng the followng equaton: T ( u) G( u) = G( x) u (4) where T: x u s the transformaton between the desgn and standard Gaussan spaces. (a) Only functon values are used (b) Functon values and local senstvtes are used Fgure 3. PDF of performance functon G(x) Torque Arm Problem 6 Amercan Insttute of Aeronautcs and Astronautcs

7 Usng local senstvty nformaton ncreases by a factor of n the number of data obtaned from each samplng pont (from one to four for the case of three desgn varables), hence the number of samplng ponts can be reduced n+ tmes. In Fgure 3(b), the PDFs of the performance functon are plotted for alternatve polynomals expansons and that obtaned usng Monte Carlo smulaton. Note that a stochastc response surface wth the same level of accuracy to that showed n Fgure 3(a) can be obtaned wth four tmes less number of samplng ponts. B. Relablty-Based Desgn Optmzaton The relablty optmzaton problem under consderaton requres to mnmze the mass of the torque arm whle satsfyng stress relablty constrants. Let the model output G be defned as G σ ( x ) = (5) σ max Usng Eq. (9), the desgn optmzaton problem can be defned as Mnmze Mass( d) subject to PG ( ( x) 0) Φ( βt), =,, NC (6) L U d d d where β t s the target relablty ndces and Φ s the cumulatve dstrbuton functon of the standard normal dstrbuton. For the relablty analyss, a target relablty ndex of 3.0 s used, whch s equvalent to 99.87% relablty. The stress values at four (.e., NC = 4) dfferent locatons are montored. Table shows the lower and upper bounds of the mean values assocated wth the desgn varables (modeled as random varables). Note that the desgn parameters are the relatve movement of the corner ponts, the ntal values for all desgn parameters s zero. The lower and upper bounds are chosen such that the topology of the boundary s preserved throughout the whole desgn process. Table. Defnton of random desgn parameters and mean value bounds Random d L d d U Standard Dstrbuton Varables (Intal) (Optmum) Devaton type d Normal d Normal d Normal d Normal d Normal d Normal d Normal d Normal d * For comparson purposes, ths RBDO problem s solved usng all random varables wthout any adaptve reducton. At each desgn pont, the eght random varables are used to construct the SRS. In order to generate the thrd-order SRS, a total of 89 samplng ponts are used; at each samplng pont stress and local senstvty nformaton s gathered. The optmzaton problem converges at the 2-th teraton. The desgn varables at the optmum desgn are lsted n the fourth column of Table, and the optmum geometry s plotted n Fgure 4(a). Fgure 4(b) shows the stress dstrbuton of the torque arm model at the optmum desgn. The maxmum stress occurs at Pont A wth a value of 704 MPa. Consderng the maxmum allowable stress lmt s 800 MPa, the mean value of the optmum desgn has about 96 MPa margn. Fgure 5 shows the desgn hstory of the cost functon. The ntal mass of kg s reduced to kg (about 59.4%) at the optmum desgn. Most reducton has been acheved n the frst fve desgn cycles, and after that the optmzaton slowly converged by adjustng desgn parameters. 7 Amercan Insttute of Aeronautcs and Astronautcs

8 (a) (b) Fgure 4. Optmum desgn and stress dstrbuton of the torque arm model wth 8 random varables. (a) Blue color = ntal desgn, black color = optmum desgn (b) Max. equvalent stress = 704 MPa at Pont A. Fgure 5. Optmzaton hstory of cost functon (mass) for the torque arm model wth 8 random varables. C. Adaptve Reducton of Random Varables The RBDO problem n the prevous secton was solved wth all random varables. However, some random varables dd not sgnfcantly contrbute to the stress functon varance. Thus, a sgnfcant amount of computatonal cost can be saved f the random varables whose contrbuton to the varance of the output s small are consdered as determnstc varables at ther mean values. Ths secton descrbes how the global senstvty ndces (man factors) can be used for decdng whether to fx unessental random varables durng the constructon of stochastc response surfaces. At the ntal desgn stage, a lower-order stochastc response surface s constructed usng all random varables. In ths partcular example the frst-order SRS s constructed usng 7 samplng ponts. At the ntal desgn, the frstorder SRS wth eght random varables can be expressed as, G = a = u + 0.7u u u u 0.052u u 0.06u (7) One useful aspect of the polynomal chaos expanson s that the coeffcents n Eq. (7) are a measure of the contrbuton of the correspondng random varable to the varaton of the output, and these coeffcents wll not change sgnfcantly n hgher-order SRS. On the other hand, typcally the man factor assocated wth a partcular varable s responsble for most of ts contrbuton to the output varance. Thus, evaluatng the global senstvty ndces (man factors) usng the frstorder SRS can be justfed. Note that all random varables are transformed nto SRV, the varance of G Fgure 6. Optmum desgns for the full SRS (black color) and adaptvely reduced SRS (blue color). Because some varables are fxed, the nteror cutout of the desgn from the adaptvely reduced SRS s larger than that from the full SRS. can be evaluated analytcally. Usng Eqs. (7) and (8), the global senstvty ndex of each random varable s calculated. Usng Eq. (7) and assumng the desgn varables are ndependent, the global senstvty ndex can be calculated as: S = a 2 n 2 a j j= (8) 8 Amercan Insttute of Aeronautcs and Astronautcs

9 If the global senstvty ndex of a specfc varable s less than a threshold value, the varable s consdered as determnstc and fxed at ts mean value. In order to show the advantage of the adaptve reducton of random varables, the torque arm problem s solved usng a threshold value of.0%. Table 2 shows the frst-order SRS of the torque arm model at the ntal desgn. The total varance of stress functon s Based on the global senstvty ndces, there are only three random varables whose GSI s greater than.0%;.e., u 2, u 6, and u 8. Thus, n the relablty analyss only these three random varables are used n constructng the thrd-order SRS, whch now requres only 9 samplng ponts. All other random varables are consdered as determnstc varables at ther mean values. If the total number of samplng ponts for both low (7) and hgher-order (9) polynomal expansons are compared wth the hgher-order SRS usng all random varables (89), a sgnfcant reducton of the number of samplng ponts was acheved. The RBDO problem, defned n Eq. (6) s now solved usng the proposed adaptve reducton of random varables. The optmzaton algorthm converges after the 7-th teraton. As seen n Fgure 6, the optmum desgn usng the adaptvely reduced SRS s slghtly dfferent from that obtaned n the prevous secton (wthout adaptve reducton). The former has a longer nteror cutout than the latter. Ths can be explaned from the fact that some varables were consdered determnstc throughout the desgn process. Furthermore, the optmum value acheved usng the adaptvely reduced SRS converges to a lower value than the one wthout adaptve reducton). The total mass of the torque arm s reduced n 57.6%. The dfference between the two approaches s approxmately.8%. The number of actve random varables assocated wth the modelng of the frst constrant durng the desgn teratons are lsted n Table 3. On average, four random varables were preserved as such, whch mples that only 29 samplng ponts were requred for constructng the SRS. Ths s three tmes less than the SRS approach wthout adaptve reducton (89 samplng ponts). Table 2. Global senstvty ndces (man factors) for the torque arm model at the ntal desgn. Only three random varables (u 2, u 6, and u 8 ) are preserved when a threshold value of.0% s n place. SRV Varance GSI (%) u u u u u u u u Table 3. Comparson of the number of random varables n each desgn cycle. The threshold of.0% s used. The frst constrant s lsted. VI. Conclusons In ths paper, we present an approach for solvng RBDO problems nvolvng a computatonally demandng model. Key aspects of the approach are: ) the uncertanty propagaton of random varables usng a polynomal chaos expanson and local senstvty nformaton, and ) the use of global senstvty nformaton to adaptvely reduced the number of random varables throughout the desgn process. The convergence and accuracy of the proposed approach was demonstrated usng a benchmark case and an ndustral relablty-based desgn optmzaton problem (automotve part). Iter Full SRS Reduced SRS VII. References. J. Tu, K. K. Cho, and Y. H. Park, 999, A New Study on Relablty Based Desgn Optmzaton, ASME Journal of Mechancal Desgn, Vol. 2(4) 2. N. H. Km, H. Wang, N. V. Quepo, 2004, Effcent shape optmzaton usng polynomal chaos expanson and local sensvtes, 9 th ASCE Specalty Conference on Probablstc Mechancs and Structural Relablty, July 25-28, Albuquerque, NM 9 Amercan Insttute of Aeronautcs and Astronautcs

10 3. I. Sobol, Senstvty Analyss for Nonlnear Mathematcal Models, Mathematcal Modelng & Computatonal Experment, 993, Vol., pp T. Homma and A. Saltell, Importance Measures n Global Senstvty Analyss of Nonlnear Models, Relablty Engneerng and System Safety, 996, Vol. 52, pp A. Saltell, S. Tarantola, and K. Chan, A Quanttatve Model-Independent Method for Global Senstvty Analyss of Model Output, Technometrcs, February 999, Vol. 4, No., pp Devroye, L. (986) Nonunform random varate generaton. New York: Sprnger Verlag. 7. S. S. Isukapall, A. Roy and P. G. Georgopoulos, Stochastc Response Surface Methods (SRSMs) for Uncertanty Propagaton: Applcaton to Envron-mental and Bologcal Systems. Rsk Analyss, 998, Vol. 8, No. 3, pp S. S. Isukapall, A. Roy and P. G. Georgopoulos, Effcent Senstvty/Uncertanty Analyss Usng the Combned Stochastc Response Surface Method and Automated Dfferentaton: Applcaton to Envronmental and Bologcal Systems, Rsk Analyss, 2000, Vol. 20, No. 5, pp Carle, A., M. Fagan, and L. L. Green (998). Prelmnary results from the applcaton of automated adjont code generaton to cfl3d. In 7th AIAA/USAF/NASA/ISSMO Symposum on Multdscplnary Analyss and Optmzaton, Volume Part 2, AIAA , pp Sept F. van Keulen, R. T. Haftka, and N. H. Km, Re-vew of Optons for Structural Desgn Senstvty Analyss. Part: Lnear Systems, Computer Methods of Appled Mechancs and Engneerng, to appear, Enevoldsen, I. and Sorensen, J.D., 994, Relablty-Based Optmzaton n Structural Engneerng, Structural Safety, Vol. 5, pp Chandu, S.V.L. and Grandh, R.V., 995, General Purpose Procedure for Relablty Based Structural Optmzaton under Parametrc Uncertantes, Advances n Engneerng Software, Vol. 23, pp Wu, Y.T. and Wang, W., 996, A New Method for Effcent Relablty-Based Desgn Optmzaton, Probablstc Mechancs & Structural Relablty: Proceedngs of the 7th Specal Conference, pp Grandh, R.V. and Wang, L.P., 998, Relablty-Based Structural Optmzaton Usng Improved Two-Pont Adaptve Nonlnear Approxmatons, Fnte Elements n Analyss and Desgn, Vol. 29, pp N.H.Km et al. Numercal method for shape optmzaton usng meshfree method. Struct Multdsc Optm 24,48-429, Sprnger-Verlag, J.Vlladsen and M.L.Mchelsen. Soluton of Dfferental Equaton Models by Polynomal Approxmaton. Prentce-Hall, Englewood Clffs, New Jersey, Amercan Insttute of Aeronautcs and Astronautcs